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Smoothness in algebraic geography. (English) Zbl 1032.16012
The most naive way to understand a finite-dimensional associative algebra is to find a basis and analyze its multiplication table. In the modern incarnation, one considers the scheme of all associative $$n$$-dimensional algebras over the field $$k$$ as a subscheme of affine space $$\operatorname{Hom}_k(k^n\otimes k^n,k^n)$$. Then $$\text{GL}_n(k)$$ acts on the $$k$$-rational points of the scheme so that orbits can be interpreted as isomorphism classes of $$n$$-dimensional algebras. So the isomorphism classes of algebras can be understood as a subvariety of $$\operatorname{Hom}_k(k^n\otimes k^n,k^n)$$, which is denoted by $$\text{Alg}_r$$. The paper analyses the smoothness of some orbits. The particular case where $$X_n$$ is the component of the matrix ring is of importance. The paper answers negatively a question of Seshadri and shows that $$X_n$$ is singular for $$n\geq 3$$. The name Algebraic Geography was given by Flanigan which referred to the study of the variety $$\text{Alg}_r$$ in this fashion.

##### MSC:
 16G10 Representations of associative Artinian rings 16R30 Trace rings and invariant theory (associative rings and algebras) 16P10 Finite rings and finite-dimensional associative algebras 16S80 Deformations of associative rings
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